Considerations on Computing Real Logarithms of Matrices, Hamiltonian Logarithms, and Skew-symmetric Logarithms

In this work, the issue of computing a real logarithm of a real matrix is addressed. After a brief review of some known methods, more attention is paid to three methods: (i) Padé approximation techniques, (ii) Newton’s method, and (iii) a series expansion method. Newton’s method has not been previously treated in the literature; we address commutativity issues, and simplify the algorithmic formulation. We also address general structure preserving issues for two applications in which we are interested: finding the real Hamiltonian logarithm of a symplectic matrix, and the skew-symmetric logarithm of an orthogonal matrix. The diagonal Padé approximants and the proposed series expansion technique are proven to be structure preserving. Some algorithmic issues are discussed. Notation and a few known Facts. A matrix M ∈ IR is called Hamiltonian if MJ + JM = 0, where J = ( 0 I −I 0 ) ; equivalently, M has the block structure M = ( A B C −AT ) , where all blocks are (n × n) and B and C are symmetric. A matrix T is called symplectic if T JT = J ; equivalently, T = −JT J , so that if T = ( A B C D ) , then T = ( D −BT −CT A ) . A matrix S ∈ IR is skew-symmetric if S = −S, and Q ∈ IR is orthogonal if QQ = I. A symplectic similarity transformation of a symplectic (Hamiltonian) matrix is symplectic (Hamiltonian). Hamiltonian and skew-symmetric matrices are closed under sum, multiplication by a scalar, transposition, and commutator operator. Symplectic and orthogonal matrices are closed under inversion, transposition, and multiplication.